Hawking and rotational fields near a bifurcate sphere

Let $(O,g)$ be a connected oriented smooth four-dimensional Ricci-flat spacetime, with the metric and Killing-field conventions of Stationary Lorentzian data and tensor conventions. Let $(S,H_1,H_2)$ be a regular bifurcate non-expanding null horizon pair in the precise pre-rigidity sense of Regular bifurcate null horizon data before rigidity. There is an open neighborhood $O^{\mathrm{\prime }}\subseteq O$ of $S$ and a nontrivial smooth Killing field $K$ tangent to the null generators of both branches. This conclusion assumes no previously existing Killing generator.

If additionally a smooth Killing field $T$ is tangent to both branches and $T{\mathrm{\mid }}_S\equiv ̸0$, one may shrink $O^{\mathrm{\prime }}$ so that $[T,K]=0$ and there is a constant ${\lambda }_0$ for which $Z=T+{\lambda }_{0}K$ is a nontrivial rotational Killing field commuting with $T$. Rotational means that its local flow has a common positive period on the chosen neighborhood; after constant rescaling it has the normalized circle-action convention of Periodic isometries and invariant normal averaging. The conclusions concern a full neighborhood of $S$, not extension over the entire exterior.

Theorem 1.1 of Alexakis–Ionescu–Klainerman supplies the full-neighborhood Hawking field under the stated regular non-expanding bifurcate-horizon hypotheses. Their Theorem 1.2 and §5 Propositions 5.1–5.2 supply the commuting field and the linear combination $Z=T+{\lambda }_{0}K$ with a common positive period under the additional branch-tangent symmetry hypothesis. These conclusions are imported inputs, not proofs reconstructed here. The five preceding local items formulate and relate the pre-rigidity data; they do not claim to replace the characteristic existence and unique-continuation arguments of the cited source. The exact source statement, hypotheses and actual reading limits are recorded in the citation metadata.